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The two closely related Lorentz-invariant partial orders of space-time aredistinguished with respect to the existence of antichain cutsets and thepossibility of grading. World lines of particles with or without mass are themaximal chains in the causality order of space-time, and antichain cutsets arethe levels of the various gradings of the causality partial order. The maximalchains of the weaker, subluminal causality order need not be connectedtopologically, subluminal causality has no antichain cutsets and cannot begraded. Combinatorial characterizations of optical lines and hyperplanes,separation lines, inertia planes and lines, ultimately in terms of thecausality order yield a simple proof of the Alexandrov-Zeeman Theorem.

Remarks on antichains in the causality order of space-time